Convex semigroups on L^p-like spaces

Abstract

In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having L^p-spaces in mind as a typical application. We show that the basic results from linear C₀-semigroup theory extend to the convex case. We prove that the generator of a convex C₀-semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup, a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of C₀-semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations.